Second-order differential equations

A second-order differential equation involves a function's second derivative, and the most important case in applications is the linear equation with constant coefficients, ay'' + by' + cy = 0, which shows up throughout physics as the equation governing everything from a mass on a spring to an RLC electrical circuit. Solving it systematically starts with a trial: suppose the solution has the form y = e^(rx) for some constant r yet to be found. This is not a guess pulled from nowhere — exponentials are the unique functions whose derivatives are proportional to themselves, so substituting this trial form turns every derivative in the equation into a power of r times the same exponential factor, which can be divided out, leaving a plain algebraic equation in r: ar² + br + c = 0, called the characteristic equation.

The nature of the characteristic equation's roots determines the entire qualitative behaviour of the solution, and there are exactly three cases. If the characteristic equation has two distinct real roots r1 and r2, the general solution is a sum of two exponentials, y = C1·e^(r1x) + C2·e^(r2x) — this is the overdamped case in a physical oscillator, where the system returns to equilibrium without oscillating. If the roots are repeated (a single real root r appearing twice), the general solution is y = (C1 + C2x)·e^(rx), where the extra factor of x is required precisely because a single exponential alone cannot supply the second independent solution needed — this is the critically damped case, the fastest possible return to equilibrium without overshooting. If the roots are a complex conjugate pair, α ± iβ, the general solution can be rewritten, using Euler's formula, in purely real terms as y = e^(αx)(C1cos(βx) + C2sin(βx)) — an oscillating solution whose amplitude grows or decays exponentially depending on the sign of α; this is the underdamped case, oscillation with an exponentially decaying envelope, and reduces to pure undamped oscillation when α = 0.

In every case, the general solution contains exactly two arbitrary constants, matching the equation's order — the same counting principle as for a first-order equation needing one constant, now doubled because specifying a unique physical solution (say, the motion of an oscillating mass) requires fixing both an initial position and an initial velocity, two independent pieces of information.